As Peter outlines heat soak comes through a series of places. I have a Series 2 and the platform/age is mostly the same.

Where the air intake around the top of the heater is open to the engine bay and not sealed hot air gets into the system and so the cabin, it can also smell a little oil/greasy as an indicator. I put some soft foam around the edge of it which pushes very gently against the underside of the bonnet which did improve the air quality.
I assume you are aware that there are two air vents on the bulkhead hidden way in front of the gear lever. I mention it as they are not obvious and you are a new poster. These are closed in general unless I have the A/Con on. They just vent through warm air and often blow directly onto your knees.
The bigger problem is general heating of the alloy chassis which then heats the frame around you in the tub. All if needs is a sealed roof (who am I kidding) and you have a little pressure cooker when pressing on!
The matting Peter points out along with some around the exhaust can improve things. Some people did add vents into the bonnet to vent the heat away quicker. If you get the factory to fit them then do ask them to check the bloke doing it can count (sorry forum joke as one came out with different number of louvres!)
The gearstick can become hot. You can replace it (original was a plastic BMW) with many designs, I used a metal ball, perhaps a mistake as it also gets hot. Just as well it pulls in every gear.

In general the engine temp sits around 90-95C on the S2 with the fans coming in around 95/98 or if the air/con is switched on.

Where are you based ?

Everyone loves a Morgan. Even me, unless it's broken again.